HIGHER-DIMENSIONAL ANALOGS OF DONALDSON-WITTEN THEORY

We present a Donaldson-Witten-type field theory in eight dimensions on manifolds with Spin(7) holonomy, We prove that the stress tensor is B RST exact for metric variations pre serving the holonomy and we give t he invariants for this class of variations, In six and seven dimension s we propose similar theories on Calabi-Yau threefolds and manifolds o f G(2) holonomy, respectively, We point out that these theories arise by considering supersymmetric Yang-Mills theory defined on such manifo lds, The theories are invariant under metric Variations preserving the holonomy structure without the need for twisting, This statement is a higher-dimensional analogue of the fact that Donaldson-Witten field t heory on hyper-Kahler 4-manifolds is topological without twisting. Hig her-dimensional analogues of Floer cohomology are briefly outlined, Al l of these theories arise naturally within the context of string theor y. (C) 1997 Elsevier Science B.V.